Question

Lim x tends to 0 Sin5x-sin3x/sinx

Answer 1

The value of given limit problem is 2.

Step-by-step explanation:

The given limit problem is

$Lim_{x\rightarrow 0}(\dfrac{\sin 5x-\sin 3x}{\sin x})$

It can be rewritten as

$Lim_{x\rightarrow 0}(\dfrac{\sin 5x}{\sin x}-\dfrac{\sin 3x}{\sin x})$

$Lim_{x\rightarrow 0}(\dfrac{\sin 5x}{\sin x}\times \dfrac{5x}{5x}-\dfrac{\sin 3x}{\sin x}\times \dfrac{3x}{3x})$

$Lim_{x\rightarrow 0}(\dfrac{\sin 5x}{5x}\times \dfrac{5x}{\sin x}-\dfrac{\sin 3x}{3x}\times \dfrac{3x}{\sin x})$

Distribute limits.

$Lim_{x\rightarrow 0}\dfrac{\sin 5x}{5x}\times 5Lim_{x\rightarrow 0}\dfrac{x}{\sin x}-Lim_{x\rightarrow 0}\dfrac{\sin 3x}{3x}\times 3Lim_{x\rightarrow 0}\dfrac{x}{\sin x}$

We know that $Lim_{x\rightarrow 0}\dfrac{\sin ax}{ax}=Lim_{x\rightarrow 0}\dfrac{ax}{\sin ax}=1$

After applying limits we get

$1\times 5(1)-1\times 3(1)$

$5-3$

$2$

Therefore, the value of given limit problem is 2.

#Learn more

Evaluate lim x-> 0 sin5x/sin3x

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